91Ó°ÊÓ

Use an appropriate coordinate system to find the volume of the given solid. The region below \(x^{2}+y^{2}+z^{2}=4 z\) and above \(z=\sqrt{x^{2}+y^{2}}\)

Evaluate the iterated integral by first changing the order of integration. $$\int_{0}^{1} \int_{\sqrt{x}}^{1} \frac{3}{4+y^{3}} d y d x$$

Sketch graphs of the cylindrical equations. $$\theta=\pi / 4$$

Sketch graphs of the cylindrical equations. $$r=4$$

Show that the volume under the cone \(z=k-r\) and above the \(x y\) -plane (where \(k>0\) ) grows as a cubic function of \(k\). Show that the volume under the paraboloid \(z=k-r^{2}\) and above the \(x y\) -plane (where \(k>0\) ) grows as a quadratic function of \(k\) Explain why this volume increases less rapidly than that of the cone.

Relate to unit basis vectors in spherical coordinates. For the position vector \(\quad \mathbf{r}=\langle x, y, z\rangle=\langle\rho \cos \theta \sin \phi\) \(\rho \sin \theta \sin \phi, \rho \cos \phi)\) in spherical coordinates, compute the unit vector \(\hat{\rho}=\frac{\mathbf{r}}{r},\) where \(r=\|\mathbf{r}\| \neq 0\)

When solving projectile motion problems, we track the motion of an object's center of mass. For a high jumper, the athlete's entire body must clear the bar. Amazingly, a high jumper can accomplish this without raising his or her center of mass above the bar. To see how, suppose the athelete's body is bent into a shape modeled by the region between \(y=\sqrt{9-x^{2}}\) and \(y=\sqrt{8-x^{2}}\) with the bar at the point \((0,2) .\) Assuming constant mass density, show that the center of mass is below the bar, but the body does not touch the bar. CANT COPY THE IMAGE

Sketch the cardioid \(r=2-2 \sin \theta\) in the \(x y\) -plane. Define \(\bar{r}\) and \(\tilde{\theta}\) to be polar coordinates in the \(y z\) -plane (that is, \(y=\tilde{r} \cos \tilde{\theta}\) and \(z=\tilde{r} \sin \tilde{\theta}) .\) Show that \(\tilde{\theta}=\frac{\pi}{2}-\phi\) and \(\cos \phi=\sin \tilde{\theta} .\) Use this information to graph \(\rho=2-2 \cos \phi\)